Extreme Value Theorem (EVT) (must be memorized for exam)
The Extreme Value Theorem (EVT) is a fundamental result in calculus
The Extreme Value Theorem (EVT) is a fundamental result in calculus. It states that for a continuous function defined on a closed interval, there must be at least one maximum value and one minimum value within that interval.
More formally, let’s consider a function f(x) that is continuous on a closed interval [a,b]. According to the Extreme Value Theorem, there must exist some values c and d within the interval [a,b] such that f(c) is the maximum value of f(x) on [a,b] and f(d) is the minimum value of f(x) on [a,b].
In simpler terms, if we have a continuous function defined on a closed interval, there will always be a highest and a lowest point on that interval. These extreme points can either occur at the endpoints of the interval (a and b) or at some point(s) within the interval.
The EVT is an important theorem because it guarantees the existence of these extreme points for continuous functions, allowing us to find the maximum and minimum values within a given interval. It is often used in calculus to find optimal solutions or analyze the behavior of functions over specific intervals.
Remembering the statement and implications of the Extreme Value Theorem is crucial as it can be applied in various optimization problems, finding critical points, and determining bounds for functions.
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